Question

1.tính

a) \(\sqrt{3-2\sqrt{2}}\)

b)\(\sqrt{28+10\sqrt{3}}\)

c)\(\sqrt{14+6\sqrt{5}}\)

d)\(\sqrt{13-4\sqrt{3}}\)

2.tính

a)\(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\)

b)\(\sqrt{18+8\sqrt{2}}-\sqrt{18-8\sqrt{2}}\)

Answer 1

1a)\(\sqrt{3-2\sqrt{2}}=\sqrt{2-2.\sqrt{2}.1+1}=\sqrt{\left(\sqrt{2}-1\right)^2}=\left|\sqrt{2}-1\right|=\sqrt{2}-1\)

b)\(\sqrt{28+10\sqrt{3}}=\sqrt{25+2.5.\sqrt{3}+3}=\sqrt{\left(5+\sqrt{3}\right)^2}=\left|5+\sqrt{3}\right|=5+\sqrt{3}\)

c)\(\sqrt{14+6\sqrt{5}}=\sqrt{9+2.3.\sqrt{5}+5}=\sqrt{\left(3+\sqrt{5}\right)^2}=\left|3+\sqrt{5}\right|=3+\sqrt{5}\)

d)\(\sqrt{13-4\sqrt{3}}=\sqrt{13-2.\sqrt{12}}=\sqrt{12-2.\sqrt{12}.1+1}=\sqrt{\left(\sqrt{12}-1\right)^2}=\sqrt{12}-1=2\sqrt{3}-1\)

2)a)\(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}=\sqrt{7-2.\sqrt{7}.1+1}-\sqrt{7+2.\sqrt{7}.1+1}=\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}=\left|\sqrt{7}-1\right|-\left|\sqrt{7}+1\right|=\sqrt{7}-1-\left(\sqrt{7}+1\right)=-2\)

b)\(\sqrt{18+8\sqrt{2}}-\sqrt{18-8\sqrt{2}}\)

\(=\sqrt{16+2.4.\sqrt{2}+2}-\sqrt{16-2.4.\sqrt{2}+2}\)

\(=\sqrt{\left(4+\sqrt{2}\right)^2}-\sqrt{\left(4-\sqrt{2}\right)^2}\)

\(=\left|4+\sqrt{2}\right|-\left|4-\sqrt{2}\right|=4+\sqrt{2}-\left(4-\sqrt{2}\right)=2\sqrt{2}\)

Answer 2

\(a,\sqrt{3-2\sqrt{2}}=\sqrt{\left(\sqrt{2}\right)^2-2\sqrt{2}+1}\)

\(=\sqrt{\left(\sqrt{2}-1\right)^2}=\left|\sqrt{2}-1\right|=\sqrt{2}-1\)\(\)

b, \(\sqrt{28+10\sqrt{3}}=\sqrt{28+2\cdot5\cdot\sqrt{3}}\)

\(=\sqrt{5^2-2\cdot5\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}\)

\(=\sqrt{\left(5-\sqrt{3}\right)^2}=\left|5-\sqrt{3}\right|=5-\sqrt{3}\)

Answer 3

\(c,\sqrt{14+6\sqrt{5}}=\sqrt{14+2\cdot3\cdot\sqrt{5}}=\sqrt{3^2+6\sqrt{5}+\left(\sqrt{5}\right)^2}=\sqrt{\left(3+\sqrt{5}\right)^2}=\left|3+\sqrt{5}\right|=3+\sqrt{5}\)

d,\(\sqrt{13-4\sqrt{3}}=\sqrt{13-2\cdot2\cdot\sqrt{3}}=\sqrt{\left(2\sqrt{3}\right)^2-4\sqrt{3}+1}=\sqrt{\left(2\sqrt{3}-1\right)^2}=\left|2\sqrt{3}-1\right|=2\sqrt{3}-1\)

Answer 4

2.

a,\(\sqrt{8-2\sqrt{7}}-\sqrt{8+2\sqrt{7}}\)

\(=\sqrt{\left(\sqrt{7}\right)^2-2\sqrt{7}+1}-\sqrt{\left(\sqrt{7}\right)^2+2\sqrt{7}+1}\)

\(=\sqrt{\left(\sqrt{7}-1\right)^2}-\sqrt{\left(\sqrt{7}+1\right)^2}\)

\(=\left|\sqrt{7}-1\right|-\left|\sqrt{7}+1\right|\)

\(=\sqrt{7}-1-\sqrt{7}-1=-2\)

b, \(\sqrt{18+8\sqrt{2}}-\sqrt{18-8\sqrt{2}}\)

\(=\sqrt{18+2\cdot4\sqrt{2}}-\sqrt{18-2\cdot4\cdot\sqrt{2}}\)

\(=\sqrt{4^2+8\sqrt{2}+\left(\sqrt{2}\right)^2}-\sqrt{4^2-8\sqrt{2}+\left(\sqrt{2}\right)^2}\)

\(=\sqrt{\left(4+\sqrt{2}\right)^2}-\sqrt{\left(4-\sqrt{2}\right)^2}\)

\(=\left|4+\sqrt{2}\right|-\left|4-\sqrt{2}\right|\)

\(=4+\sqrt{2}-4+\sqrt{2}=2\sqrt{2}\)